Replicated Transfer Matrix Analysis of Ising Spin Models on `Small World' Lattices

TL;DR

This paper develops a theoretical framework using replica symmetry to analyze Ising models on small world lattices, deriving phase diagrams and effective field distributions for both ferromagnetic and spin-glass cases.

Contribution

It introduces a replica symmetric approach to analyze Ising models on small world networks with quenched disorder, deriving phase diagrams and field distributions.

Findings

Derived phase diagrams for small world Ising models.

Calculated effective field distributions for different bond types.

Extended the transfer matrix method to disordered small world lattices.

Abstract

We calculate equilibrium solutions for Ising spin models on `small world' lattices, which are constructed by super-imposing random and sparse Poissonian graphs with finite average connectivity c onto a one-dimensional ring. The nearest neighbour bonds along the ring are ferromagnetic, whereas those corresponding to the Poisonnian graph are allowed to be random. Our models thus generally contain quenched connectivity and bond disorder. Within the replica formalism, calculating the disorder-averaged free energy requires the diagonalization of replicated transfer matrices. In addition to developing the general replica symmetric theory, we derive phase diagrams and calculate effective field distributions for two specific cases: that of uniform sparse long-range bonds (i.e. `small world' magnets), and that of (+J/-J) random sparse long-range bonds (i.e. `small world' spin-glasses).